## Ideal TheoryIdeal theory is important not only for the intrinsic interest and purity of its logical structure but because it is a necessary tool in many branches of mathematics. In this introduction to the modern theory of ideals, Professor Northcott assumes a sound background of mathematical theory but no previous knowledge of modern algebra. After a discussion of elementary ring theory, he deals with the properties of Noetherian rings and the algebraic and analytical theories of local rings. In order to give some idea of deeper applications of this theory the author has woven into the connected algebraic theory those results which play outstanding roles in the geometric applications. |

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### Contents

RESIDUE RINGS AND RINGS OF QUOTIENTS | 31 |

the algebraic theory | 63 |

Continuation of 4 3 | 81 |

Notes | 101 |

110 | |

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algebraic geometry analytically independent assume called Cauchy sequence choose coefficients completes the proof completion of Q composition series consequently Corollary cr(a definition denote dim Q dimQ elements of Q extension ring find an integer finite number form of degree Further hence homomorphism ideal theory ideal which contains integral domain integrally closed intersection isomorphism Lemma Let Q mapping maximal ideal minimal base minimal prime ideal modulo multiplicatively closed Noetherian ring normal decomposition obtain ordp polynomial ring power series primary decomposition primary ideal primary ring prime ideal belonging proper ideal proper prime ideal Proposition 9 prove q is p.primary Q is regular rank regular local ring relevant prime ideal residue class residue rings result ring and let ring of quotients ring Q sequence in Q sequence of elements shows suppose system of parameters Theorem unit element zero divisor zero element zero ideal